An infinite product challenge

Gil Kalai wrote a blog post yesterday entitled “Test Your Intuition (or knowledge, or programming skills) 36.” The challenge is to evaluate the infinite product

\prod_{p\,\, \mathrm{prime}} \frac{p^2+1}{p^2 - 1}

I imagine there’s an elegant analytical solution, but since the title suggested that programming might suffice, I decided to try a little Python. I used primerange from SymPy to generate the list of primes up to 200, and cumprod from NumPy to generate the list of partial products.

        [(p*p+1)/(p*p-1) for p in primerange(1,200)]

Apparently the product converges to 5/2, and a plot suggests that it converges very quickly.

Plot of partial products

Here’s another plot to look more closely at the rate of convergence. Here we look at the difference between 5/2 and the partial products, on a log scale, for primes less than 2000.

Plot of 2.5 minus partial products, log scale


One thought on “An infinite product challenge

  1. (brief proof sketch) Rearrange the terms so that they are in terms of powers of 1/p. Then notice you can multiply the top and the bottom by (1-(1/p^2)). The answer becomes recognizable from the product formula for the Riemann zeta function as \zeta(2)^2 / zeta(4) = 90/36 = 5/2.

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